### Correlation coefficient: the last block of statistical foundation

Correlation has already been mentioned in

Statistical measures and their geometric roots

Properties of standard deviation

The pearls of AP Statistics 35

The pearls of AP Statistics 33

### The hierarchy of definitions

Suppose random variables are not constant. Then their standard deviations are not zero and we can define their correlation as in Chart 1.

### Properties of correlation

**Property 1**. *Range of the correlation coefficient*: for any

This follows from the **Cauchy-Schwarz inequality**, as explained here.

Recall from this post that correlation is cosine of the angle between

**Property 2**. *Interpretation of extreme cases*. (Part 1) If

(Part 2) If

**Proof**. (Part 1)

(1)

which, in turn, implies that

Plugging this in Eq. (1) we get

The proof of Part 2 is left as an exercise.

**Property 3**. Suppose we want to measure correlation between weight *The correlation coefficient is unit-free* in the sense that it does not depend on the units used: *correlation is homogeneous of degree in both arguments.*

**Proof**. One measurement is proportional to another,

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